Abstract
This article focuses on developing and analyzing an efficient numerical scheme for solving two-dimensional singularly perturbed parabolic convection–diffusion initial-boundary value problems exhibiting a regular boundary layer. For approximating the time derivative, we use the Peaceman–Rachford alternating direction implicit method on uniform mesh and for the spatial discretization, a hybrid finite difference scheme is proposed on a special rectangular mesh which is tensor-product of piecewise-uniform Shishkin meshes in the spatial directions. We prove that the numerical scheme converges uniformly with respect to the perturbation parameter \(\varepsilon \) and also attains almost second-order spatial accuracy in the discrete supremum norm. Finally, numerical results are presented to validate the theoretical results. In addition to this, numerical experiments are conducted to demonstrate the effect of the time-dependent boundary conditions in the order of convergence numerically by introducing the classical evaluation of the boundary data; and also the improvement in the spatial order of accuracy of the present method by considering the Bakhvalov–Shishkin mesh in the spatial directions.
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The authors wish to acknowledge the referees for their valuable comments and suggestions, which helped to improve the presentation.
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Mukherjee, K., Natesan, S. Parameter-uniform fractional step hybrid numerical scheme for 2D singularly perturbed parabolic convection–diffusion problems. J. Appl. Math. Comput. 60, 51–86 (2019). https://doi.org/10.1007/s12190-018-1203-y
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DOI: https://doi.org/10.1007/s12190-018-1203-y
Keywords
- Singularly perturbed parabolic problem
- Regular boundary layer
- Numerical scheme
- Alternating directions
- Piecewise-uniform Shishkin mesh
- Bakhvalov–Shishkin mesh uniform convergence